{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Realization of Recursive Filters\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quantization of Variables and Operations\n",
    "\n",
    "As for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-Effects), the practical realization of recursive filters may suffer from the quantization of variables and algebraic operations. The effects of [coefficient quantization](quantization_of_coefficients.ipynb) were already discussed. This section takes a look at the quantization of variables.  We limit the investigations to the recursive part of a second-order section (SOS), since any recursive filter of order $N \\geq 2$ can be [decomposed into SOSs](cascaded_structures.ipynb).\n",
    "\n",
    "The computation of the output signal $y[k] = \\mathcal{H}\\{ x[k] \\}$ by a difference equation involves a number of multiplications and additions. As discussed already for [non-recursive filters](../nonrecursive_filters/quantization_effects.ipynb#Quantization-of-Signals-and-Operations), multiplying two numbers in a binary representation (e.g. [two's complement](https://en.wikipedia.org/wiki/Two's_complement) or [floating point](https://en.wikipedia.org/wiki/Floating_point)) requires requantization of the result to keep the word length constant. The addition of two numbers may fall outside the maximum/minimum values of the representation and may suffer from clipping.\n",
    "\n",
    "The resulting round-off and clipping errors depend on the number and sequence of algebraic operations. These depend on the structure used for implementation of the SOSs. For ease of illustration we limit our discussion to the [direct form I and II](direct_forms.ipynb). Similar insights can be achieved in a similar manner for other structures."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Analysis of Round-Off Errors\n",
    "\n",
    "Round-off errors are a consequence of reducing the word length after a multiplication. In order to investigate the influence of these errors on a recursive filter, the statistical model for [round-off errors in multipliers](../nonrecursive_filters/quantization_effects.ipynb#Model-for-round-off-errors-in-multipliers) as introduced for non-recursive filters is used. We furthermore neglect clipping. \n",
    "\n",
    "The difference equation for the recursive part of a SOS realized in direct form I or II is given as\n",
    "\n",
    "\\begin{equation}\n",
    "y[k] =  x[k] - a_1 \\, y[k-1] - a_2 \\, y[k-2]\n",
    "\\end{equation}\n",
    "\n",
    "where $a_0 = 1$, $a_1$ and $a_2$ denote the coefficients of the recursive part. Introducing the requantization after the multipliers into the difference equation yields the output signal $y_Q[k]$\n",
    "\n",
    "\\begin{equation}\n",
    "y_Q[k] = x[k] - \\mathcal{Q} \\{ a_{1} \\, y[k-1] \\} - \\mathcal{Q} \\{ a_{2} \\, y[k-2] \\}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\mathcal{Q} \\{ \\cdot \\}$ denotes the requantizer. Requantization is a non-linear process which results in a requantization error. If the value to be requantized is much larger that the quantization step $Q$, the average statistical properties of this error can be modeled as additive uncorrelated white noise. Introducing the error into above difference equation gives\n",
    "\n",
    "\\begin{equation}\n",
    "y_Q[k] = x[k] - a_1 \\, y[k-1] - e_1[k] - a_2 \\, y[k-2] - e_2[k]\n",
    "\\end{equation}\n",
    "\n",
    "where the two white noise sources $e_1[k]$ and $e_2[k]$ are assumed to be uncorrelated to each other. This difference equation can be split into a set of two difference equations\n",
    "\n",
    "\\begin{align}\n",
    "y_Q[k] &= y[k] + e[k] \\\\\n",
    "y[k] &=  x[k] - a_1 \\, y[k-1] - a_2 \\, y[k-2] \\\\\n",
    "e[k] &= - e_1[k] - e_2[k] - a_1 \\, e[k-1] - a_2 \\, e[k-2]\n",
    "\\end{align}\n",
    "\n",
    "The first difference equation computes the desired output signal $y[k]$ as a result of the input signal $x[k]$. The second one the additive error $e[k]$ due to requantization as a result of the requantization error $- (e_1[k] + e_2[k])$ injected into the recursive filter.\n",
    "The power spectral density (PSD) $\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ of the error $e[k]$ is then given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = | H(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})|^2 \\cdot (\\Phi_{e_1 e_1}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) + \\Phi_{e_2 e_2}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}))\n",
    "\\end{equation}\n",
    "\n",
    "According to the model for the requantization errors, their PSDs are given as $\\Phi_{e_1 e_1}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\Phi_{e_2 e_2}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{Q^2}{12}$. Introducing this together with the transfer function of the SOS yields\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\left| \\frac{1}{1 + a_1 \\, \\mathrm{e}^{\\,-\\mathrm{j}\\,\\Omega} + a_2 \\, \\mathrm{e}^{\\,-\\mathrm{j}\\,2\\,\\Omega}} \\right|^2 \\cdot \\frac{Q^2}{6}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Round-off error of a SOS\n",
    "\n",
    "The following example evaluates the error $e[k] = y_Q[k] - y[k]$ for a SOS which only consists of a recursive part. The desired system response $y[k]$ is computed numerically by floating point operations with double precision, $y_Q[k]$ is computed by applying a uniform midtread quantizer after the multiplications. The system is excited by uniformly distributed white noise. Besides the PSD $\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$, the signal-to-noise ratio (SNR) $10 \\cdot \\log_{10} \\left( \\frac{\\sigma_y^2}{\\sigma_e^2} \\right)$ in dB of the filter is evaluated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "SNR due to requantization: 45.470119 dB\n"
     ]
    },
    {
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kZSAvTGVuZ3RoIDg4ID4+CnN0cmVhbQp4nD2NuRHAMAgEc6qgBIF4pH48juT+U5/wEwA77NxcunJjTaxUx0w+hCRm4UUao2iRSmPrDeT4PRQar/3CsKPzdjasrqOhW4Jk2kfIbVe9669bdN7JCxvOCmVuZHN0cmVhbQplbmRvYmoKMTcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzMzMgPj4Kc3RyZWFtCnicPZFLbsQwDEP3OYUuUMD6x+eZoqv2/ts+OdMuAhGUYpFU3y1LQuVD7ZaO5jP51MvrcPJzefZBnk518dinvi6PfDPrqW7vjr0ZrS1T9JZK+Cpwzz699xS7FRrgnpJLHLoRFAG9Afs+f2RQEZDbaOh2KXSab6neYkz2iiNJpW2L78dJ5D71dc0Lgwotrcl/S6pGbJvCruM+/UkhmKYDclS1LbGaefSzDyMEskcWK1CZJT07Mp8gv6+If/gXJSRzVRNqLDvIM3Byi5NS2ei0VlwW3giEeVsTzOjUOyUm0AmIqBVFvse1TnLsVpa61qnW4+AgvCkalENOcmqTuVnKHTLrcOV+CgHCAeZ9jEaH6Bo+51nITB5qKoczm+W1QMRcTBjCy+pck46D+jlVEMCpSKUDSs5aiEgs5Rz02TMxB4fWOAH9Rfa6vn4Bnzh/NgplbmRzdHJlYW0KZW5kb2JqCjE4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzI3ID4+CnN0cmVhbQp4nD2SS47EMAhE9zkFF2jJ/Pw5T0az6rn/th92ehZRoQBFUbhrSpPo8lKXHCrZm/zoFVPFzeTviYbENHE30MEp97WjCP4EuA5m7gzRSolBvXWJyvKRsfPX19OXB+/L22GwUXqWmFefkrEGC/J0dPEB5tza7n+V7yuTjiy9Sk9F70u9WJaoDzH7YvUp3MbmG9t6MJ5M8Ws2UauZU9baIroM1EVKN7HVC1DtUYHjXEdvFW0DEIE7QQeVCbVqbMFISLRF1i/MaUL3xP/tAbpf2sR17IPYtI1nfEUaQ9DduEdZw8wx6nxqc8Pa09km65TiTC7XsjhWCTRKy1hqS/Csel0UY4MZaKUQzdFqbGVodujrZXg/yAxvO4qxqKVKz543p+BcfezXFHrwPIsdwZuc2C3o55CsQgYtNmHqLF2v8Dnnff1+ACo9fUYKZW5kc3RyZWFtCmVuZG9iagoxOSAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE1MiA+PgpzdHJlYW0KeJw9T8sRQyEIvFvFNsCMIILW8zI5mf6vAU08sbK4H/GBCmIWkFSHuMOt4sWFtaOr41OkHbQKi4PmBpa/ErGCvIHr1ukYPWabeIrnxhg6Y4awcMyRDFnaxTPuti9Fjg2Cu2FWoekplj7kemEbB1J6s7RdoaZwLqYiY4Zx22mfWyXSjtMvSv2ariK9w9SStYvGn31/Abz3NmcKZW5kc3RyZWFtCmVuZG9iagoxNCAwIG9iago8PCAvQmFzZUZvbnQgL0RlamFWdVNhbnMtT2JsaXF1ZSAvQ2hhclByb2NzIDE1IDAgUgovRW5jb2RpbmcgPDwgL0RpZmZlcmVuY2VzIFsgNzIgL0ggODEgL1EgMTAxIC9lIDEwNiAvaiBdIC9UeXBlIC9FbmNvZGluZyA+PgovRmlyc3RDaGFyIDAgL0ZvbnRCQm94IFsgLTEwMTYgLTM1MSAxNjYwIDEwNjggXSAvRm9udERlc2NyaXB0b3IgMTMgMCBSCi9Gb250TWF0cml4IFsgMC4wMDEgMCAwIDAuMDAxIDAgMCBdIC9MYXN0Q2hhciAyNTUgL05hbWUgL0RlamFWdVNhbnMtT2JsaXF1ZQovU3VidHlwZSAvVHlwZTMgL1R5cGUgL0ZvbnQgL1dpZHRocyAxMiAwIFIgPj4KZW5kb2JqCjEzIDAgb2JqCjw8IC9Bc2NlbnQgOTI5IC9DYXBIZWlnaHQgMCAvRGVzY2VudCAtMjM2IC9GbGFncyA5NgovRm9udEJCb3ggWyAtMTAxNiAtMzUxIDE2NjAgMTA2OCBdIC9Gb250TmFtZSAvRGVqYVZ1U2Fucy1PYmxpcXVlCi9JdGFsaWNBbmdsZSAwIC9NYXhXaWR0aCAxMzUwIC9TdGVtViAwIC9UeXBlIC9Gb250RGVzY3JpcHRvciAvWEhlaWdodCAwID4+CmVuZG9iagoxMiAwIG9iagpbIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwCjYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgNjAwIDYwMCA2MDAgMzE4IDQwMSA0NjAgODM4IDYzNgo5NTAgNzgwIDI3NSAzOTAgMzkwIDUwMCA4MzggMzE4IDM2MSAzMTggMzM3IDYzNiA2MzYgNjM2IDYzNiA2MzYgNjM2IDYzNiA2MzYKNjM2IDYzNiAzMzcgMzM3IDgzOCA4MzggODM4IDUzMSAxMDAwIDY4NCA2ODYgNjk4IDc3MCA2MzIgNTc1IDc3NSA3NTIgMjk1CjI5NSA2NTYgNTU3IDg2MyA3NDggNzg3IDYwMyA3ODcgNjk1IDYzNSA2MTEgNzMyIDY4NCA5ODkgNjg1IDYxMSA2ODUgMzkwIDMzNwozOTAgODM4IDUwMCA1MDAgNjEzIDYzNSA1NTAgNjM1IDYxNSAzNTIgNjM1IDYzNCAyNzggMjc4IDU3OSAyNzggOTc0IDYzNCA2MTIKNjM1IDYzNSA0MTEgNTIxIDM5MiA2MzQgNTkyIDgxOCA1OTIgNTkyIDUyNSA2MzYgMzM3IDYzNiA4MzggNjAwIDYzNiA2MDAgMzE4CjM1MiA1MTggMTAwMCA1MDAgNTAwIDUwMCAxMzUwIDYzNSA0MDAgMTA3MCA2MDAgNjg1IDYwMCA2MDAgMzE4IDMxOCA1MTggNTE4CjU5MCA1MDAgMTAwMCA1MDAgMTAwMCA1MjEgNDAwIDEwMjggNjAwIDUyNSA2MTEgMzE4IDQwMSA2MzYgNjM2IDYzNiA2MzYgMzM3CjUwMCA1MDAgMTAwMCA0NzEgNjE3IDgzOCAzNjEgMTAwMCA1MDAgNTAwIDgzOCA0MDEgNDAxIDUwMCA2MzYgNjM2IDMxOCA1MDAKNDAxIDQ3MSA2MTcgOTY5IDk2OSA5NjkgNTMxIDY4NCA2ODQgNjg0IDY4NCA2ODQgNjg0IDk3NCA2OTggNjMyIDYzMiA2MzIgNjMyCjI5NSAyOTUgMjk1IDI5NSA3NzUgNzQ4IDc4NyA3ODcgNzg3IDc4NyA3ODcgODM4IDc4NyA3MzIgNzMyIDczMiA3MzIgNjExIDYwOAo2MzAgNjEzIDYxMyA2MTMgNjEzIDYxMyA2MTMgOTk1IDU1MCA2MTUgNjE1IDYxNSA2MTUgMjc4IDI3OCAyNzggMjc4IDYxMiA2MzQKNjEyIDYxMiA2MTIgNjEyIDYxMiA4MzggNjEyIDYzNCA2MzQgNjM0IDYzNCA1OTIgNjM1IDU5MiBdCmVuZG9iagoxNSAwIG9iago8PCAvSCAxNiAwIFIgL1EgMTcgMCBSIC9lIDE4IDAgUiAvaiAxOSAwIFIgPj4KZW5kb2JqCjI0IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTYwID4+CnN0cmVhbQp4nD2QSxLDIAxD95xCR8D4f550ukruv60NnWywGIT0wH1iIqMWJ4KvxIcGpcMk8GwVhHsw2RFyhOQETYaygyRxDQ0GpcKmY5nvyaZ10kqsvNF+2Xc0Zp10ii1Fh3Yh/+f1Ityj4BrrGTwPYDHI2kql7JkwWmWmrvKo6LUf0+E9D0SrxeX1Bq7w2kc3qSgqS+r+RHc02P6Rt/se3x+ZZTgkCmVuZHN0cmVhbQplbmRvYmoKMjUgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA3OSA+PgpzdHJlYW0KeJxNzbsNwCAMBNCeKTwC4P8+UaqwfxsbIkJjP+lOOsEOFdzisBhod7ha8aVRmH3qmRKSUHM9RFgzJTqEpF/6yzDDmNjItu+3Vu4X3hscGQplbmRzdHJlYW0KZW5kb2JqCjI2IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIzNwovU3VidHlwZSAvRm9ybSAvVHlwZSAvWE9iamVjdCA+PgpzdHJlYW0KeJw9UbtxxTAM6z0FRuBH/Gied5cq2b8NKDkpeIApEQTkpyzRhZ9niOD7We7/yAOSrVBthCc0FZEN08DnSRFYbqQm3F7c54RslSP24lwgwhDtPAlppAsWOxkL3hc/j6seZqy5Yfy+M5p9VHTVUR28ew7jZk0/TpTd682sjlub+3TvrhOHa0gmn/cfnJRKp5csgzpLuLA2mhrW47woxljMOP4nqrBNsrajCsHSJUgq0IAYShLGgMUt/iInWg4L2psbaeudyU6qNIqGF6MM3qD1RjiKdJF8mGsrg7GpmDa++eQlN+j7Z7+fr18Da1rrCmVuZHN0cmVhbQplbmRvYmoKMjcgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNjUgPj4Kc3RyZWFtCnicRY87EgMhDEN7TqEjgH/AeTaTir1/G8s7SRosjCU/ois69srDY2PKxmu0sSfCFu5SOg2nqYyviqdnXaDLYTJTb1zNXGCqsMhuTrH6GHyh8uzmhK9VnhjCl0wJDTCVO7mH9fpRnJZ8JLsLguqUjcrCMEfS90BMTZunhYH8jy95akFQmeaNa5aVR2sVUzRnmCpbC4L1gaA6pfoD0/9Mp70/3PQ9gAplbmRzdHJlYW0KZW5kb2JqCjI4IDAgb2JqCjw8IC9CQm94IFsgLTEwMjEgLTQ2MyAxNzk0IDEyMzMgXSAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDIzOAovU3VidHlwZSAvRm9ybSAvVHlwZSAvWE9iamVjdCA+PgpzdHJlYW0KeJxFkE2ywzAIg/c+BUcwmB/7PJ15q/T+20okfV1kYAxI+jKWh0QeeQ877EwsUPFq5hJa8hq6VbyOaE5x211XOifs5hRumJ++MVVMqKJ7CVU1Q+ijcTD5Ol7DffXbe/ix3oplfRX51cmZrZxW7dS1vdkxDTeYjjd3XqqQgKokog8dX/+OV6eoIjZsN9xIMcHo4IuF3Q3jyahi1eW2ReNhgmmY9XpgjF9xEFLxC6CX7g2coQ9w2Rf413F6jTB0DLiBovjWeRQLkehRS9uS9c7AziqFG8zImzs0VYhBVVC1YT3g88GeP/xr/H0AqH5ggAplbmRzdHJlYW0KZW5kb2JqCjI5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzM4ID4+CnN0cmVhbQp4nDVSS5JbQQjbv1PoAq5q/s15nJrV5P7bCOysoIEWEpAWOMjESwxRjXLFH3mC8TqBv+vlafw+3oXUgqci/cC1aRvvx5o1UbA0YinMPvb9KCHHU+PfEOi5SBNmZDJyIBmI+7U+f9abTDn8BqRpc/ooSXoQLdjdGnZ8WZBB0pMaluzkh3UtsLoITZgbayIZObUyNc/HnuEynhgjQdUsIEmfuE8VjEgzHjtnLXmQ4XiqFy9+vY3XMo+pl1UFMrYJ5mA7mQmnKCIQv6AkuYm7aOoojmbGmtuFhpIi9909nJz0ur+cRAVeCeEs1hKOGXrKMic7DUqgauUEmGG99oVxmjZKuFPT7V2xr99nJmHc5rCzUjINznFwL5vMESR73TFhEx6HmPfuEYzEvPldbBFcucy5JtOP/SjaSB8U1+dcTZmtKOEfquSJFdf4//zez88/kDd9sQplbmRzdHJlYW0KZW5kb2JqCjMwIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzA0ID4+CnN0cmVhbQp4nD2SO5LDMAxDe52CF8iM+JPk82Qnlff+7T4yyVaASYkAKC91mbKmPCBpJgn/0eHhYjvld9iezczAtUQvE8spz6ErxNxF+bKZjbqyOsWqwzCdW/SonIuGTZOa5ypLGbcLnsO1ieeWfcQPNzSoB3WNS8IN3dVoWQrNcHX/O71H2Xc1PBebVOrUF48XURXm+SFPoofpSuJ8PCghXHswRhYS5FPRQI6zXK3yXkL2DrcassJBaknnsyc82HV6Ty5uF80QD2S5VPhOUezt0DO+7EoJPRK24VjufTuasekamzjsfu9G1sqMrmghfshXJ+slYNxTJkUSZE62WG6L1Z7uoSimc4ZzGSDq2YqGUuZiV6t/DDtvLC/ZLMiUzAsyRqdNnjh4yH6NmvR5led4/QFs83M7CmVuZHN0cmVhbQplbmRvYmoKMzEgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA1NCA+PgpzdHJlYW0KeJwzNjZXMFAwNDJX0DUyNlUwMjRQMDczUUgx5IIxc8EssGwOF1whhAmSz4GrzOFKAwBMkA8VCmVuZHN0cmVhbQplbmRvYmoKMzIgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyMzAgPj4Kc3RyZWFtCnicNVFJbsMwDLzrFfOBAOIuv8dBT+3/rx3SCWBgaEuczREbGxF4icHPQeTGW9aMmvibyV3xuzwVHgm3gidRBF6Ge9kJLm8Yl/04zHzwXlo5kxpPMiAX2fTwRMhgl0DowOwa1GGbaSf6hoTPjkg1G1lOX0vQS6sQKE/ZfqcLSrSt6s/tsy607WtPONntqSeVTyCeW7ICl41XTBZjGfRE5S7F9EGqs4WehPKifA6y+aghEl2inIEnBgejQDuw57afiVeFoHV1n7aNoRopHU//NjQ1SSLkEyWc2dK4W/j+nnv9/AOmVFOfCmVuZHN0cmVhbQplbmRvYmoKMzMgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyMjcgPj4Kc3RyZWFtCnicNU87sgMhDOs5hS6QGYxtYM+zmVQv92+fZLINEv5I8vRERyZe5sgIrNnxthYZiBn4FlPxrz3tw4TqPbiHCOXiQphhJJw167ibp+PFv13lM9bBuw2+YpYXBLYwk/WVxZnLdsFYGidxTrIbY9dEbGNd6+kU1hFMKAMhne0wJcgcFSl9sqOMOTpO5InnYqrFLr/vYX3BpjGiwhxXBU/QZFCWPe8moB0X9N/Vjd9JNIteAjKRYGGdJObOWU741WtHx1GLIjEnpBnkMhHSnK5iCqEJxTo7CioVBZfqc8rdPv9oXVtNCmVuZHN0cmVhbQplbmRvYmoKMzQgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNDUgPj4Kc3RyZWFtCnicRVC7jUMxDOs9BRcIYP0se553SJXbvz1KRnCFIVo/kloSmIjASwyxlG/iR0ZBPQu/F4XiM8TPF4VBzoSkQJz1GRCZeIbaRm7odnDOvMMzjDkCF8VacKbTmfZc2OScBycQzm2U8YxCuklUFXFUn3FM8aqyz43XgaW1bLPTkewhjYRLSSUml35TKv+0KVsq6NpFE7BI5IGTTTThLD9DkmLMoJRR9zC1jvRxspFHddDJ2Zw5LZnZ7qftTHwPWCaZUeUpnecyPiep81xOfe6zHdHkoqVV+5z93pGW8iK126HV6VclUZmN1aeQuDz/jJ/x/gOOoFk+CmVuZHN0cmVhbQplbmRvYmoKMzUgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAzOTIgPj4Kc3RyZWFtCnicPVJLbgUxCNvPKbhApfBNcp6p3u7df1ubzFSqCi8DtjGUlwypJT/qkogzTH71cl3iUfK9bGpn5iHuLjam+FhyX7qG2HLRmmKxTxzJL8i0VFihVt2jQ/GFKBMPAC3ggQXhvhz/8ReowdewhXLDe2QCYErUbkDGQ9EZSFlBEWH7kRXopFCvbOHvKCBX1KyFoXRiiA2WACm+qw2JmKjZoIeElZKqHdLxjKTwW8FdiWFQW1vbBHhm0BDZ3pGNETPt0RlxWRFrPz3po1EytVEZD01nfPHdMlLz0RXopNLI3cpDZ89CJ2Ak5kmY53Aj4Z7bQQsx9HGvlk9s95gpVpHwBTvKAQO9/d6Sjc974CyMXNvsTCfw0WmnHBOtvh5i/YM/bEubXMcrh0UUqLwoCH7XQRNxfFjF92SjRHe0AdYjE9VoJRAMEsLO7TDyeMZ52d4VtOb0RGijRB7UjhE9KLLF5ZwVsKf8rM2xHJ4PJntvtI+UzMyohBXUdnqots9jHdR3nvv6/AEuAKEZCmVuZHN0cmVhbQplbmRvYmoKMzYgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxMzMgPj4Kc3RyZWFtCnicTY9BEsMwCAPvfoWegLEB8550ekr+fy2QNu4F7YyAkYYwCDxiDOswJbx6++FVpEtwNo75JRlFPAhqC9wXVAVHY4qd+Njdoeyl4ukUTYvrEXPTtKR0N1Eqbb2dyPjAfZ/eH1W2JJ2CHlvqhC7RJPJFAnPYVDDP6sZLS4+n7dneH2Y+M9cKZW5kc3RyZWFtCmVuZG9iagozNyAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDI0NyA+PgpzdHJlYW0KeJxNUbttRDEM698UXOAA62t5ngtSXfZvQ8kIkMIgoS8ppyUW9sZLDOEHWw++5JFVQ38ePzHsMyw9yeTUP+a5yVQUvhWqm5hQF2Lh/WgEvBZ0LyIrygffj2UMc8734KMQl2AmNGCsb0kmF9W8M2TCiaGOw0GbVBh3TRQsrhXNM8jtVjeyOrMgbHglE+LGAEQE2ReQzWCjjLGVkMVyHqgKkgVaYNfpG1GLgiuU1gl0otbEuszgq+f2djdDL/LgqLp4fQzrS7DC6KV7LHyuQh/M9Ew7d0kjvfCmExFmDwVSmZ2RlTo9Yn23QP+fZSv4+8nP8/0LFShcKgplbmRzdHJlYW0KZW5kb2JqCjM4IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggOTAgPj4Kc3RyZWFtCnicTY1BEsAgCAPvvCJPUETQ/3R60v9fq9QOvcBOAokWRYL0NWpLMO64MhVrUCmYlJfAVTBcC9ruosr+MklMnYbTe7cDg7LxcYPSSfv2cXoAq/16Bt0P0hwiWAplbmRzdHJlYW0KZW5kb2JqCjM5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNjggPj4Kc3RyZWFtCnicMzK3UDBQsDQBEoYWJgrmZgYKKYZcQL6piblCLhdIDMTKAbMMgLQlnIKIW0I0QZSCWBClZiZmEEk4AyKXBgDJtBXlCmVuZHN0cmVhbQplbmRvYmoKNDAgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAyNTUgPj4Kc3RyZWFtCnicRZFLkgMgCET3noIjgPzkPJmaVXL/7TSYTDZ2l6j9hEojphIs5xR5MP3I8s1ktum1HKudjQKKIhTM5Cr0WIHVnSnizLVEtfWxMnLc6R2D4g3nrpxUsrhRxjqqOhU4pufK+qru/Lgsyr4jhzIFbNY5DjZw5bZhjBOjzVZ3h/tEkKeTqaPidpBs+IOTxr7K1RW4Tjb76iUYB4J+oQlM8k2gdYZA4+YpenIJ9vFxu/NAsLe8CaRsCOTIEIwOQbtOrn9x6/ze/zrDnefaDFeOd/E7TGu74y8xyYq5gEXuFNTzPRet6wwd78mZY3LTfUPnXLDL3UGmz/wf6/cPUIpmiAplbmRzdHJlYW0KZW5kb2JqCjQxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTYxID4+CnN0cmVhbQp4nEWQSxLDIAxD95xCR/BHBnyedLpK77+tIU2zgKexQAZ3JwSptQUT0QUvbUu6Cz5bCc7GeOg2bjUS5AR1gFak42iUUn25xWmVdPFoNnMrC60THWYOepSjGaAQOhXe7aLkcqbuzvlHcPVf9Uex7pzNxMBk5Q6EZvUp7nybHVFd3WR/0mNu1mt/FfaqsLSspeWE285dM6AE7qkc7f0FqXM6hAplbmRzdHJlYW0KZW5kb2JqCjQyIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjE0ID4+CnN0cmVhbQp4nD1QuxFDMQjrPQUL5M587TfPy6XL/m0knKRCNkISlJpMyZSHOsqSrClPHT5LYoe8h+VuZDYlKkUvk7Al99AK8X2J5hT33dWWs0M0l2g5fgszKqobHdNLNppwKhO6oNzDM/oNbXQDVocesVsg0KRg17YgcscPGAzBmROLIgxKTQb/rXL3UtzvPRxvooiUdPCu+eX0y88tvE49jkS6vfmKa3GmOgpEcEZq8op0YcWyyEOk1QQ1PQNrtQCu3nr5N2hHdBmA7BOJ4zSlHEP/1rjH6wOHilL0CmVuZHN0cmVhbQplbmRvYmoKNDMgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA4MCA+PgpzdHJlYW0KeJxFjLsNwDAIRHumYAR+JmafKJWzfxsgStxwT7p7uDoSMlPeYYaHBJ4MLIZT8QaZo2A1uEZSjZ3so7BuX3WB5npTq/X3BypPdnZxPc3LGfQKZW5kc3RyZWFtCmVuZG9iago0NCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE0NyA+PgpzdHJlYW0KeJw9T7kNAzEM6z0FFzjAeixb81yQ6rJ/G8pGUggiQPGRZUfHClxiApOOORIvaT/4aRqBWAY1R/SEimFY4G6SAg+DLEpXni1eDJHaQl1I+NYQ3q1MZKI8rxE7cCcXowc+VBtZHnpAO0QVWa5Jw1jVVl1qnbACHLLOwnU9zKoE5dEnaykfUFRCvXT/n3va+wsAby/rCmVuZHN0cmVhbQplbmRvYmoKNDUgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCAxNDkgPj4Kc3RyZWFtCnicNY9LDgMhDEP3OYUvMFJ+hHAeqq6m9982YVoJCQvbL8EWg5GMS0xg7Jhj4SVUT60+JCOPukk5EKlQNwRPaEwMM2zSJfDKdN8ynlu8nFbqgk5I5OmsNhqijGZew9FTzgqb/svcJGplRpkDMutUtxOysmAF5gW1PPcz7qhc6ISHncqw6E4xotxmRhp+/9v0/gJ7MjBjCmVuZHN0cmVhbQplbmRvYmoKNDYgMCBvYmoKPDwgL0ZpbHRlciAvRmxhdGVEZWNvZGUgL0xlbmd0aCA0OSA+PgpzdHJlYW0KeJwzNrRQMFAwNDAHkkaGQJaRiUKKIRdIAMTM5YIJ5oBZBkAaojgHriaHKw0AxugNJgplbmRzdHJlYW0KZW5kb2JqCjQ3IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjM3ID4+CnN0cmVhbQp4nE1ROW4EMQzr/Qp9YADrtOc9G2w1+X8bUt4EqURDFEnJ5SlTMuWyuSQjJGvKlw6NLbZcvg9CU0tFwdS9RXXJa5imrFssXdDzqSyv4Rjge3c31D/0iNkCkdGCXWGBDpA7uGD4PXsmbFMLIlEl1AxgmrDCHK5EDEEGY50ZBqUKg1P1d5Xjsw07BdYOZlOkR1ITnXSD5oW33nIhgq1Tuak30oTc2acYYmXjvkqX4wPgYKLLRGTE5mU4ng5haPDONGCFsx7EBJnWkdKLmZExDTfpyNVXi4rPNhlLntH/9of6K59u/4MQfMb7B0lVXH4KZW5kc3RyZWFtCmVuZG9iago0OCAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE1NyA+PgpzdHJlYW0KeJxFkLkRQzEIRHNVQQkSsAjqscfRd/+pF/lKtG8ALYevJVOqHyciptzXaPQweQ6fTSVWLNgmtpMachsWQUoxmHhOMaujt6GZh9TruKiquHVmldNpy8rFf/NoVzOTPcI16ifwTej4nzy0qehboK8LlH1AtTidSVAxfa9igaOcdn8inBjgPhlHmSkjcWJuCuz3GQBmvle4xuMF3QE3eQplbmRzdHJlYW0KZW5kb2JqCjQ5IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzMyID4+CnN0cmVhbQp4nC1SOY4kMQzL/Qp+YADr8vGeHkzU+/90SVUFBapsyzzkcsNEJX4skNtRa+LXRmagwvCvq8yF70jbyDqIa8hFXMmWwmdELOQxxDzEgu/b+Bke+azMybMHxi/Z9xlW7KkJy0LGizO0wyqOwyrIsWDrIqp7eFOkw6kk2OOL/z7FcxeCFr4jaMAv+eerI3i+pEXaPWbbtFsPlmlHlRSWg+1pzsvkS+ssV8fj+SDZ3hU7QmpXgKIwd8Z5Lo4ybWVEa2Fng6TGxfbm2I+lBF3oxmWkOAL5mSrCA0qazGyiIP7I6SGnMhCmrulKJ7dRFXfqyVyzubydSTJb90WKzRTO68KZ9XeYMqvNO3mWE6VORfgZe7YEDZ3j6tlrmYVGtznBKyV8NnZ6cvK9mlkPyalISBXTugpOo8gUS9iW+JqKmtLUy/Dfl/cZf/8BM+J8AQplbmRzdHJlYW0KZW5kb2JqCjUwIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzE3ID4+CnN0cmVhbQp4nDVSS3JDMQjbv1Nwgc6Yv32edLJq7r+thCcrsC1AQi4vWdJLftQl26XD5Fcf9yWxQj6P7ZrMUsX3FrMUzy2vR88Rty0KBFETPfgyJxUi1M/U6Dp4YZc+A68QTikWeAeTAAav4V94lE6DwDsbMt4Rk5EaECTBmkuLTUiUPUn8K+X1pJU0dH4mK3P5e3KpFGqjyQgVIFi52AekKykeJBM9iUiycr03VojekFeSx2clJhkQ3SaxTbTA49yVtISZmEIF5liA1XSzuvocTFjjsITxKmEW1YNNnjWphGa0jmNkw3j3wkyJhYbDElCbfZUJqpeP09wJI6ZHTXbtwrJbNu8hRKP5MyyUwccoJAGHTmMkCtKwgBGBOb2wir3mCzkWwIhlnZosDG1oJbt6joXA0JyzpWHG157X8/4HRVt7owplbmRzdHJlYW0KZW5kb2JqCjUxIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggNTIgPj4Kc3RyZWFtCnicMzY2VzAAQl1LIwVjINvcyFIhxZDLyNQEzMzlggnmcFkYg1XlcBlAaZiiHK40AN+qDa0KZW5kc3RyZWFtCmVuZG9iago1MiAwIG9iago8PCAvRmlsdGVyIC9GbGF0ZURlY29kZSAvTGVuZ3RoIDE3ID4+CnN0cmVhbQp4nDM2tFAwgMMUQy4AGpQC7AplbmRzdHJlYW0KZW5kb2JqCjUzIDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMTMxID4+CnN0cmVhbQp4nEWPyw0EIQxD71ThEvIZPqmH1Z7Y/q/rMJpBQvhBIjvxMAis8/I20MXw0aLDN/421atjlSwfunpSVg/pkIe88hVQaTBRxIVZTB1DYc6YysiWMrcb4bZNg6xslVStg3Y8Bg+2p2WrCH6pbWHqLPEMwlVeuMcNP5BLrXe9Vb5/QlMwlwplbmRzdHJlYW0KZW5kb2JqCjU0IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMzM4ID4+CnN0cmVhbQp4nDVSOa7dQAzrfQpdIIB2zZznBal+7t+GlF8KQ7RWipqOFpVp+WUhVS2TLr/tSW2JG/L3yQqJE5JXJdqlDJFQ+TyFVL9ny7y+1pwRIEuVCpOTksclC/4Ml94uHOdjaz+PI3c9emBVjIQSAcsUE6NrWTq7w5qN/DymAT/iEXKuWLccYxVIDbpx2hXvQ/N5yBogZpiWigpdVokWfkHxoEetffdYVFgg0e0cSXCMjVCRgHaB2kgMObMWu6gv+lmUmAl07Ysi7qLAEknMnGJdOvoPPnQsqL8248uvjkr6SCtrTNp3o0lpzCKTrpdFbzdvfT24QPMuyn9ezSBBU9YoaXzQqp1jKJoZZYV3HJoMNMcch8wTPIczEpT0fSh+X0smuiiRPw4NoX9fHqOMnAZvAXPRn7aKAxfx2WGvHGCF0sWa5H1AKhN6YPr/1/h5/vwDHLaAVAplbmRzdHJlYW0KZW5kb2JqCjU1IDAgb2JqCjw8IC9GaWx0ZXIgL0ZsYXRlRGVjb2RlIC9MZW5ndGggMjQ4ID4+CnN0cmVhbQp4nC1ROZIDQQjL5xV6QnPT77HLkff/6QrKAYOGQyA6LXFQxk8Qlive8shVtOHvmRjBd8Gh38p1GxY5EBVI0hhUTahdvB69B3YcZgLzpDUsgxnrAz9jCjd6cXhMxtntdRk1BHvXa09mUDIrF3HJxAVTddjImcNPpowL7VzPDci5EdZlGKSblcaMhCNNIVJIoeomqTNBkASjq1GjjRz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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.signal as sig\n",
    "%matplotlib inline\n",
    "\n",
    "\n",
    "N = 8192  # length of signals\n",
    "w = 8  # wordlength for requantization of multiplications\n",
    "\n",
    "\n",
    "def uniform_midtread_quantizer(x):\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x/Q + 1/2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "def no_quantizer(x):\n",
    "\n",
    "    return x\n",
    "\n",
    "\n",
    "def sos_df1(x, a, requantize=None):\n",
    "    y = np.zeros(len(x)+2)  # initial value appended\n",
    "    for k in range(len(x)):\n",
    "        y[k] = x[k] - requantize(a[1]*y[k-1]) - requantize(a[2]*y[k-2])\n",
    "\n",
    "    return y[0:-2]\n",
    "\n",
    "\n",
    "# cofficients of the SOS\n",
    "p = 0.90*np.array([np.exp(1j*np.pi/3), np.exp(-1j*np.pi/3)])\n",
    "a = np.poly(p)\n",
    "# quantization step\n",
    "Q = 1/(2**(w-1))\n",
    "\n",
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=N)\n",
    "# compute output signals w and w/o requantization\n",
    "yQ = sos_df1(x, a, requantize=uniform_midtread_quantizer)\n",
    "y = sos_df1(x, a, requantize=no_quantizer)\n",
    "# compute requantization error\n",
    "e = yQ-y\n",
    "# Signal-to-noise ratio\n",
    "SNR = 10*np.log10(np.var(y)/np.var(e))\n",
    "print('SNR due to requantization: %f dB' % SNR)\n",
    "\n",
    "# estimate PSD of requantization error\n",
    "nf, Pxx = sig.welch(e, window='hamming', nperseg=256, noverlap=128)\n",
    "Pxx = .5*Pxx  # due to normalization in scipy.signal\n",
    "Om = 2*np.pi*nf\n",
    "# compute frequency response of system\n",
    "w, H = sig.freqz([1, 0, 0], a)\n",
    "\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.plot(Om, Pxx/Q**2 * 12, 'b', label=r'$|\\hat{\\Phi}_{ee}(e^{j \\Omega})|$')\n",
    "plt.plot(w, np.abs(H)**2 * 2, 'g', label=r'$|H(e^{j \\Omega})|^2$')\n",
    "plt.title('Estimated PSD and transfer function of requantization noise')\n",
    "plt.xlabel(r'$\\Omega$')\n",
    "plt.ylabel(r'$Q^2/12$')\n",
    "plt.axis([0, np.pi, 0, 100])\n",
    "plt.legend()\n",
    "plt.grid()"
   ]
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   "source": [
    "### Small Limit Cycles\n",
    "\n",
    "Besides the requantization noise, recursive filters may be subject to periodic oscillations present at the output. These undesired oscillations are termed *limit cycles*. Small limit cycles emerge from the additive round-off noise due to requantization after a multiplication. The feedback in a recursive filter leads to a feedback of the requantization noise. This may lead to a periodic output signal with an amplitude range of some quantization steps $Q$, even after the input signal is zero. The presence, amplitude and frequency of small limit cycles depends on the location of poles and the structure of the filter. A detailed treatment of this phenomenon is beyond the scope of this notebook and can be found in the literature."
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   "source": [
    "#### Example - Small limit cycles of a SOS\n",
    "\n",
    "The following example illustrates small limit cycles for the system investigated in the previous example. The input signal is uniformly distributed white noise till time-index $k=256$ and zero for the remainder."
   ]
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   "source": [
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=256)\n",
    "x = np.concatenate((x, np.zeros(1024)))\n",
    "# compute output signal\n",
    "yQ = sos_df1(x, a, requantize=uniform_midtread_quantizer)\n",
    "\n",
    "# plot results\n",
    "np.seterr(divide='ignore')\n",
    "plt.figure(figsize=(10, 3))\n",
    "plt.plot(20*np.log10(np.abs(yQ)))\n",
    "plt.title('Level of output signal')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$|y_Q[k]|$ in dB')\n",
    "plt.grid()\n",
    "\n",
    "plt.figure(figsize=(10, 3))\n",
    "k = np.arange(1000, 1050)\n",
    "plt.stem(k, yQ[k]/Q, use_line_collection=True)\n",
    "plt.title('Output signal for zero input')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$y_Q[k] / Q$ ')\n",
    "plt.axis([k[0], k[-1], -3, 3])\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* Estimate the period of the small limit cycles. How is it related to the poles of the system?\n",
    "* What amplitude range is spanned?\n",
    "\n",
    "Solution: The period of the small limit cycles can be estimated from the second illustration as $P = 6$. The normalized frequency of a harmonic exponential signal with the same periodicity is given as $\\Omega_0 = \\frac{2 \\pi}{P} = \\frac{\\pi}{3}$. The poles of the system can be extracted from the code of the first example as $z_{\\infty 0,1} = 0.9 \\cdot e^{\\pm j \\frac{\\pi}{3}}$. The periodicity of the small limit cycles is hence linked to the normalized frequency of the poles. The amplitude range spanned by the small limit cycles is $\\pm 2 Q$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Large Limit Cycles\n",
    "\n",
    "Large limit cycles are periodic oscillations of a recursive filter due to overflows in the multiplications/additions. As for small limit cycles, large limit cycles may be present even after the input signal is zero. Their level is typically in the range of the minimum/maximum value of the requantizer. Large limit cycles should therefore be avoided in a practical implementation. The presence of large limit cycles depends on the scaling of input signal and coefficients, as well as the strategy used to cope for clipping. Amongst others, they can be avoided by proper scaling of the coefficients to prevent overflow. Again, a detailed treatment of this phenomenon is beyond the scope of this notebook and can be found in the literature."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Example - Large limit cycles of a SOS\n",
    "\n",
    "The following example illustrates large limit cycles for the system investigated in the first example. In order to trigger large limit cycles, the coefficients of the filter have been doubled. The input signal is uniformly distributed white noise till time-index $k=256$ and zero for the remainder."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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   "source": [
    "def uniform_midtread_quantizer(x, xmin=1):\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    if x <= -xmin:\n",
    "        x = -1\n",
    "    if x > xmin - Q:\n",
    "        x = 1 - Q\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x/Q + 1/2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "# compute input signal\n",
    "x = np.random.uniform(low=-1, high=1, size=256)\n",
    "x = np.concatenate((x, np.zeros(1024)))\n",
    "# compute output signal\n",
    "yQ = sos_df1(x, 2*a, requantize=uniform_midtread_quantizer)\n",
    "\n",
    "# plot results\n",
    "plt.figure(figsize=(10, 3))\n",
    "plt.plot(20*np.log10(np.abs(yQ)))\n",
    "plt.title('Level of output signal')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$|y_Q[k]|$ in dB')\n",
    "plt.grid()\n",
    "\n",
    "plt.figure(figsize=(10, 3))\n",
    "k = np.arange(1000, 1050)\n",
    "plt.stem(k, yQ[k], use_line_collection=True)\n",
    "plt.title('Output signal for zero input')\n",
    "plt.xlabel(r'$k$')\n",
    "plt.ylabel(r'$y_Q[k]$ ')\n",
    "plt.grid()"
   ]
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   "source": [
    "**Exercise**\n",
    "\n",
    "* Determine the period of the large limit cycles. How is it related to the poles of the system?\n",
    "\n",
    "Solution: The period of the large limit cycles can be estimated from the second illustration as $P = 6$. The normalized frequency of a harmonic exponential signal with the same periodicity is given as $\\Omega_0 = \\frac{2 \\pi}{P} = \\frac{\\pi}{3}$. The poles of the system can be extracted from the code of the first example as $z_{\\infty 0,1} = 0.9 \\cdot e^{\\pm j \\frac{\\pi}{3}}$. The periodicity of the large limit cycles is hence linked to the normalized frequency of the poles."
   ]
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   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
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